A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in each degree. My question is: does the converse hold? that is, is every simplicial set, having a finite number of simplicies in each degree, necessarily finite?

Let G be a finite group viewed as a one object category. Then the nerve BG is a simplicial set with finitely many simplices in each dimension but it is not finite. 


Take $X$ to be the “infinitedimensional dunce’s cap”, with a unique nondegenerate simplex $x_n$ in each dimension, and with every face of $x_n$ equal to $x_{n1}$. Explicitly, $X_n = \coprod_{m \leq n} \mathrm{Surj}([n],[m])$. So it’s clear that this has finitely many simplices in each dimension, but infinitely many nondegenerate ones in total. 

